Factorization of heavy-to-light form factors in soft-colline(3)

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections

Table1:TerminologyforthevariousmomentummodesrelevanttoexclusiveBde-

cays.Themomentumcomponentsaregivenas(n+p,p⊥,n p),butmassdimension

hastoberestoredbymultiplyingappropriatefactorsofmb.Twodi erenttermi-

nologiesforthesamemomentummodesareusedintheliterature.Inphysicalunits

λisoforder(Λ/mb)1/2,whereΛisthestrong-interactionscale.

MomentumscalingTerminologyII

([20],thiswork)

(1,1,1)

soft

collinear

(λ2,λ2,λ2)

(1,λ2,λ4)hardsoftcollinear

yetbeenachieved.Acomplicationwaspointedoutin[15],whereitwasshownthatsuper ciallysub-leadinginteractionsin1/mbcontributetotheB→πformfactorsatleadingpower.Someofthepreviousarguments[16,17]tojustifyorextendsomeaspectsof(1)mustthereforeberevised.Theformfactorshavebeenreconsideredin[15]intheframeworkofsoft-collineare ective eldtheory(SCET)[16,18]andthestructureoftheformula(1)wasseentoemerge.However,in[15]atechnicalde nitionof“factorizable”and“non-factorizable”termshasbeenadoptedthatdoesnotcorrespondtotheusualnotions,sothattheissuesofendpointsingularitiesandconvergenceofconvolutionintegralscouldnotbeclari ed.BelowweextendSCETintheposition-spaceformulation[17,19]tocoverthecaseofexclusivedecays,wheretheexternalcollinearlineshaveinvariantmassoforderΛ2asappropriateforexclusivedecays.Toobtainthefactorizationformula(1)wematchSCETtoane ectivetheoryfromwhichshort-distancecollinearmodeswithvirtualitymbΛareremoved.Thispointwas rstaddressedin[20]andtheformalismhasbeenworkedouttotheextentthatafactorizationtheoremforB→γlνwasestablished[8].¯(p) ,Themomentummodesrelevanttothefactorizationofformfactors M(p′)|q¯Γb|BwhereMisalightmesonoraphotonwithmomentumofordermb,andΓisaDiracmatrix,aresummarizedinTable1.Ingeneralwedecomposeamomentumas

p=(n+p)µnµ

2,(2)

22wherenµ±aretwolight-likevectors,n+=n =0withn+n =2.Thereferencedirections

n±arechosensuchthattheenergeticmasslessexternallineshaven+pofordermb.AsindicatedinTable1twodi erentterminologieshavebeenusedintheliteraturewhichhas